A category is the collection of arrows AND objects. The objects alone do not define the category, so there can be many categories with the same set of objects.
There is an additional requirement that there be an "identity" arrow from each object to itself. This and composition make arrows act enough like functions to be useful. Marshall On Mon, Apr 2, 2012 at 5:30 PM, Raul Miller <rauldmil...@gmail.com> wrote: > On Mon, Apr 2, 2012 at 4:35 PM, Marshall Lochbaum <mwlochb...@gmail.com> > wrote: > > A category is a formal thing, which means that it is literally composed > of > > "objects" and "arrows," and that these names, and the entities > themselves, > > mean nothing. > > Ok, but that is not meaningful. > > > We can think of a category as a set of objects, and for each > > ordered pair of (not necessarily distinct) objects a set of > > arrows from the first to the second. > > This seems to be missing some of the constraints that define > categories. Or is it really the case that arrows can represent > (for example) relations which are not functions? > > Function: > 0 -> 1 > 1 -> 0 > > Relation: > 0 -> 1 > 1 -> 0 > 1 -> 1 > > If arrows in categories can represent non-functions (and > apparently they can, https://en.wikipedia.org/wiki/Category_theory > says "However it is important to note that the objects of a > category need not be sets nor the arrows functions") then > we need to explicitly state which arrows represent functions > and which do not. > > Also, if there is just one category for a given set of objects > then I think that that means that we have to have an arrow > for each member of the cartesian product of all pairs of objects > as well as numerous other arrows for the subsets of objects. > > It seems to me that if there is only one category for a given set > of objects that this must be the arrows which represent functions > when the objects are 0 and 1: > > -> > 0 -> 1 > 0 -> 0 > 1 -> 1 > 1 -> 0 > 0 1 -> 0 0 > 0 1 -> 0 1 > 0 1 -> 1 0 > 0 1 -> 1 1 > > In the above, the values on the left of the arrow > represent the domain and the values on the right > of the arrow represent the corresponding codomain. > > If arrows can represent relations and if a given > set of objects has only one category then the > category with objects 0 and 1 must also contain > these arrows: > > 0 0 -> 0 1 > 1 1 -> 0 1 > 0 0 1 -> 0 1 0 > 0 0 1 -> 0 1 1 > 0 1 1 -> 0 0 1 > 0 1 1 -> 1 0 1 > 0 0 1 1 -> 0 1 0 1 > > (Again, domain is on the left and codomain is on the right.) > > In other words, if there is only one category for a given > set of objects then the cardinality of arrows must be > greater than the cardinality of objects. > > > Additionally, there is a binary operator on arrows, composition, > > which gives an arrow X->Z for each pair of arrows X->Y, Y->Z. > > Yes. > > > Categories are useful because they allow a lot of > > interpretations, or in other words many things can be > > represented as categories. The category of sets, for example, has > > an object for each set and an arrow for each function. > > It's not clear to me that sets are valid objects. One > issue is that some set definitions are paradoxes. Can > paradoxes be objects? Godel incompleteness can pose > a different kind of challenge (the set of all sets of > axioms for the non-trivial second order language L). > > > The category of rings uses rings and ring homomorphisms, etc. > > With an uncountable infinity of rings and thus a higher > uncountable infinity of arrows. Still, this is more constrained > than sets. > > > There is also a category of categories, which contains an object for each > > category (although some are barred to avoid paradoxes), and an arrow for > > each functor between categories. This allows us to work with categories > > under functors as we would any other object. > > This sounds like an extremely high order infinity just for variations > in objects that make up each category. And of course higher order > infinities for the arrows involved. > > If nothing else, a person should be very precise when specifying > an arrow. > > -- > Raul > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
